Properties

Label 2-6025-1.1-c1-0-82
Degree $2$
Conductor $6025$
Sign $-1$
Analytic cond. $48.1098$
Root an. cond. $6.93612$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.863·2-s − 1.78·3-s − 1.25·4-s + 1.53·6-s − 4.46·7-s + 2.81·8-s + 0.179·9-s − 4.87·11-s + 2.23·12-s − 4.21·13-s + 3.85·14-s + 0.0812·16-s + 4.42·17-s − 0.155·18-s − 7.05·19-s + 7.96·21-s + 4.21·22-s − 6.18·23-s − 5.01·24-s + 3.63·26-s + 5.02·27-s + 5.60·28-s + 7.05·29-s + 4.29·31-s − 5.69·32-s + 8.69·33-s − 3.82·34-s + ⋯
L(s)  = 1  − 0.610·2-s − 1.02·3-s − 0.627·4-s + 0.628·6-s − 1.68·7-s + 0.993·8-s + 0.0598·9-s − 1.47·11-s + 0.645·12-s − 1.16·13-s + 1.03·14-s + 0.0203·16-s + 1.07·17-s − 0.0365·18-s − 1.61·19-s + 1.73·21-s + 0.898·22-s − 1.28·23-s − 1.02·24-s + 0.713·26-s + 0.967·27-s + 1.05·28-s + 1.30·29-s + 0.771·31-s − 1.00·32-s + 1.51·33-s − 0.655·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-1$
Analytic conductor: \(48.1098\)
Root analytic conductor: \(6.93612\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 - T \)
good2 \( 1 + 0.863T + 2T^{2} \)
3 \( 1 + 1.78T + 3T^{2} \)
7 \( 1 + 4.46T + 7T^{2} \)
11 \( 1 + 4.87T + 11T^{2} \)
13 \( 1 + 4.21T + 13T^{2} \)
17 \( 1 - 4.42T + 17T^{2} \)
19 \( 1 + 7.05T + 19T^{2} \)
23 \( 1 + 6.18T + 23T^{2} \)
29 \( 1 - 7.05T + 29T^{2} \)
31 \( 1 - 4.29T + 31T^{2} \)
37 \( 1 - 2.88T + 37T^{2} \)
41 \( 1 - 8.91T + 41T^{2} \)
43 \( 1 + 1.06T + 43T^{2} \)
47 \( 1 + 9.89T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 9.41T + 59T^{2} \)
61 \( 1 - 4.32T + 61T^{2} \)
67 \( 1 - 0.904T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + 7.12T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 - 6.12T + 89T^{2} \)
97 \( 1 + 2.18T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.023223414364032997634015842396, −6.91545649074271437378951058332, −6.29045799618453617812996468126, −5.65088294595298000920230407889, −4.91991014245730336669869830064, −4.24673267128685509732673411095, −3.10736795338143070047570432645, −2.35579887695312022712027636684, −0.62251088924846565022089346055, 0, 0.62251088924846565022089346055, 2.35579887695312022712027636684, 3.10736795338143070047570432645, 4.24673267128685509732673411095, 4.91991014245730336669869830064, 5.65088294595298000920230407889, 6.29045799618453617812996468126, 6.91545649074271437378951058332, 8.023223414364032997634015842396

Graph of the $Z$-function along the critical line